library(tidyverse)
# ggplot() +
# stat_function(fun = function(x1) ___, aes(color = "Isocost: $3000")) +
# ylim(0, 300) +
# xlim(0, 300)5.4 Cost Minimization
Profit Maximization and Cost minimization are dual problems. While profit maximization addresses the producer’s problem directly by considering the output price, cost minimization focuses on finding the least costly way to achieve a target output level. Cost minimization also gives us powerful insights into firm decision-making and market outcomes.
The Cost Minimization Problem
Cost minimization is about finding the cheapest way to produce a desired level of output. Mathematically, we can express this as:
\[\begin{align} \min \ & w_1x_1 + w_2x_2 \\ \text{subject to} \ & f(x_1, x_2) = y \end{align}\]
Where:
- \(w_1\) and \(w_2\) are the prices of two factors of production (inputs)
- \(x_1\) and \(x_2\) are the quantities of these factors used
- \(f(x_1, x_2)\) is the production function
- \(y\) is the desired level of output
Isocosts
Isocost curves are all the bundles of inputs that would cost the same amount. If input 1 costs $15 per unit and input 2 costs $10 per unit, we have:
\[\text{Total Cost} = 15 x_1 + 10 x_2\]- Set total cost to be 3000 and solve for \(x_2\) (the variable on the y-axis). Then visualize this isocost curve using stat_function.
Answer:
\[\begin{align} \end{align}\]
Isocosts are straight lines with a slope of input price ratio \(\frac{w_1}{w_2}\). This should remind you of the consumer’s budget line!
- Add two more isocost lines to your plot from question 1: one representing all input bundles that cost $4,000, and another representing all input bundles that cost $5,000.
Answer:
Isoquants
Recall that isoquants are all the input bundles that can be used to produce the same level of output.
If our production function is the Cobb-Douglas function \(f(x_1, x_2) = x_1^{0.6} x_2^{0.4}\), then level curves of this production function are the isoquants.
- Set the production function equal to an output of 200 units, solve for \(x_2\) (the variable on the y-axis), and visualize the isoquant using
stat_function.
Answer:
- Visualize your isoquant curve for an output of 200 units on the same plot as your isocost curves from question 2.
Answer:
- Interpret the plot you made in question 4.
- Is there any way to produce 200 units of output (a point on the isoquant curve), while spending just $3000 (also a point on the $3000 isocost curve)?
Answer:
- Is there any way to produce 200 units of output (a point on the isoquant curve), while spending just $4000 (also a point on the $4000 isocost curve)?
Answer:
- Is there any way to produce 200 units of output (a point on the isoquant curve), while spending just $5000 (also a point on the $5000 isocost curve)?
Answer:
The cost-minimizing way to produce 200 units of output is where the isoquant for 200 units is just tangent to the lowest possible isocost curve. At this point of tangency, the slopes of the isoquant and isocost curves are equal.
This gives us a tangency condition (just like in consumer theory)! The slope of the isoquant is the Marginal Rate of Technical Substitution (MRTS), while the slope of the isocost curve is the input price ratio. The tangency condition is:
\[\frac{MP_{x_1}}{MP_{x_2}} = \frac{w_1}{w_2}\] Where \(MP\) refers to the marginal product of the input and \(w\) refers to that input’s price.
Marginal Returns to Inputs
Rearranging the tangency condition, we also have:
\[\frac{MP_{x_1}}{w_1} = \frac{MP_{x_2}}{w_2}\] To interpret this equation: suppose \(x_1\) refers to labor and \(x_2\) refers to capital equipment. Then \(MP_{x_1}\) refers to the marginal output per worker and \(w_1\) is the wage per worker. The “per worker” cancels, and the left hand side refers to the output per dollar spent on an extra worker, or the return to hiring an extra worker. The same way, the right hand side refers to the retun to renting an extra unit of capital. The firm solves its cost minimization problem when an extra dollar spent on workers has the same return as an extra dollar spent on capital.
This is actually obvious: if an extra dollar spent on workers earns a higher return than an extra dollar spent on capital so that \(\frac{MP_{x_1}}{w_1} \gt \frac{MP_{x_2}}{w_2}\), then the producer can produce more at a lower cost by spending more on workers. When the producer is doing the best they can, the return on an extra dollar spent on any input is equivalent.
Practice with Cost Minimization
A firm uses labor (\(L\)) and capital (\(K\)) to produce output (\(Q\)) according to the production function: \(Q = L^{0.7} K^{0.3}\). The wage rate (w) is $20 per unit of labor, and the rental rate of capital (r) is $12 per unit of capital.
- Use the tangency condition to show that the optimal ratio of L to K is \(L = 1.4 K\).
Answer:
\[\begin{align} \end{align}\]
- If the firm wants to produce 300 units of output, find the optimal quantities of L and K.
Answer:
\[\begin{align} \end{align}\]
- Calculate the total cost of production for 300 units of output.
\[\begin{align} \end{align}\]
- Let’s verify that the Cobb-Douglas shortcut works here. According to the Cobb-Douglas shortcut, what fraction of total cost should be spent on labor? On capital?
\[\begin{align} \end{align}\]
- Use the Cobb-Douglas shortcut to show that \(L^* = 0.035 TC\) and \(K^* = 0.025 TC\).
\[\begin{align} \end{align}\]
- Continuing from part e, eliminate \(TC\) to find the relationship between L and K: \(L^* = 1.4 K^*\).
\[\begin{align} \end{align}\]